diff options
author | Andres Erbsen <andreser@mit.edu> | 2017-06-15 12:12:47 -0400 |
---|---|---|
committer | Andres Erbsen <andreser@mit.edu> | 2017-06-15 12:12:47 -0400 |
commit | a36568d1d73aff5d7accc79fd28be672882f9c17 (patch) | |
tree | f6046789fbee1be1527b4caa58c8c331db4dc89f /src/Curves/Edwards/XYZT.v | |
parent | b319173e40cb219ab3b9b80e967b264e699851ad (diff) |
Edwards coordinates precomputed addition formula
Diffstat (limited to 'src/Curves/Edwards/XYZT.v')
-rw-r--r-- | src/Curves/Edwards/XYZT.v | 134 |
1 files changed, 0 insertions, 134 deletions
diff --git a/src/Curves/Edwards/XYZT.v b/src/Curves/Edwards/XYZT.v deleted file mode 100644 index 3604e9b2e..000000000 --- a/src/Curves/Edwards/XYZT.v +++ /dev/null @@ -1,134 +0,0 @@ -Require Import Coq.Classes.Morphisms. - -Require Import Crypto.Spec.CompleteEdwardsCurve Crypto.Curves.Edwards.AffineProofs. - -Require Import Crypto.Util.Notations Crypto.Util.GlobalSettings. -Require Export Crypto.Util.FixCoqMistakes. -Require Import Crypto.Util.Decidable. -Require Import Crypto.Util.Tactics.DestructHead. -Require Import Crypto.Util.Tactics.UniquePose. - -Module Extended. - Section ExtendedCoordinates. - Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} - {field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} - {char_ge_3 : @Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos BinNat.N.two)} - {Feq_dec:DecidableRel Feq}. - Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope. - Local Notation "0" := Fzero. Local Notation "1" := Fone. - Local Infix "+" := Fadd. Local Infix "*" := Fmul. - Local Infix "-" := Fsub. Local Infix "/" := Fdiv. - Local Notation "x ^ 2" := (x*x). - - Context {a d: F} - {nonzero_a : a <> 0} - {square_a : exists sqrt_a, sqrt_a^2 = a} - {nonsquare_d : forall x, x^2 <> d}. - Local Notation Epoint := (@E.point F Feq Fone Fadd Fmul a d). - - Local Notation onCurve x y := (a*x^2 + y^2 = 1 + d*x^2*y^2) (only parsing). - (** [Extended.point] represents a point on an elliptic curve using extended projective - * Edwards coordinates 1 (see <https://eprint.iacr.org/2008/522.pdf>). *) - Definition point := { P | let '(X,Y,Z,T) := P in - a * X^2*Z^2 + Y^2*Z^2 = (Z^2)^2 + d * X^2 * Y^2 - /\ X * Y = Z * T - /\ Z <> 0 }. - Definition coordinates (P:point) : F*F*F*F := proj1_sig P. - Definition eq (P1 P2:point) := - let '(X1, Y1, Z1, _) := coordinates P1 in - let '(X2, Y2, Z2, _) := coordinates P2 in - Z2*X1 = Z1*X2 /\ Z2*Y1 = Z1*Y2. - - Ltac t_step := - match goal with - | |- Proper _ _ => intro - | _ => progress intros - | _ => progress destruct_head' prod - | _ => progress destruct_head' @E.point - | _ => progress destruct_head' point - | _ => progress destruct_head' and - | _ => progress cbv [eq CompleteEdwardsCurve.E.eq E.eq E.zero E.add E.opp fst snd coordinates E.coordinates proj1_sig] in * - | |- _ /\ _ => split | |- _ <-> _ => split - end. - Ltac t := repeat t_step; Field.fsatz. - - Global Instance Equivalence_eq : Equivalence eq. - Proof using Feq_dec field nonzero_a. split; repeat intro; t. Qed. - Global Instance DecidableRel_eq : Decidable.DecidableRel eq. - Proof. intros P Q; destruct P as [ [ [ [ ] ? ] ? ] ?], Q as [ [ [ [ ] ? ] ? ] ? ]; exact _. Defined. - - Program Definition from_twisted (P:Epoint) : point := - let xy := E.coordinates P in (fst xy, snd xy, 1, fst xy * snd xy). - Next Obligation. t. Qed. - Global Instance Proper_from_twisted : Proper (E.eq==>eq) from_twisted. - Proof using Type. cbv [from_twisted]; t. Qed. - - Program Definition to_twisted (P:point) : Epoint := - let XYZT := coordinates P in let T := snd XYZT in - let XYZ := fst XYZT in let Z := snd XYZ in - let XY := fst XYZ in let Y := snd XY in - let X := fst XY in - let iZ := Finv Z in ((X*iZ), (Y*iZ)). - Next Obligation. t. Qed. - Global Instance Proper_to_twisted : Proper (eq==>E.eq) to_twisted. - Proof using Type. cbv [to_twisted]; t. Qed. - - Lemma to_twisted_from_twisted P : E.eq (to_twisted (from_twisted P)) P. - Proof using Type. cbv [to_twisted from_twisted]; t. Qed. - Lemma from_twisted_to_twisted P : eq (from_twisted (to_twisted P)) P. - Proof using Type. cbv [to_twisted from_twisted]; t. Qed. - - Program Definition zero : point := (0, 1, 1, 0). - Next Obligation. t. Qed. - - Program Definition opp P : point := - match coordinates P return F*F*F*F with (X,Y,Z,T) => (Fopp X, Y, Z, Fopp T) end. - Next Obligation. t. Qed. - - Section TwistMinusOne. - Context {a_eq_minus1:a = Fopp 1} {twice_d} {k_eq_2d:twice_d = d+d}. - Program Definition m1add - (P1 P2:point) : point := - match coordinates P1, coordinates P2 return F*F*F*F with - (X1, Y1, Z1, T1), (X2, Y2, Z2, T2) => - let A := (Y1-X1)*(Y2-X2) in - let B := (Y1+X1)*(Y2+X2) in - let C := T1*twice_d*T2 in - let D := Z1*(Z2+Z2) in - let E := B-A in - let F := D-C in - let G := D+C in - let H := B+A in - let X3 := E*F in - let Y3 := G*H in - let T3 := E*H in - let Z3 := F*G in - (X3, Y3, Z3, T3) - end. - Next Obligation. - match goal with - | [ |- match (let (_, _) := coordinates ?P1 in let (_, _) := _ in let (_, _) := _ in let (_, _) := coordinates ?P2 in _) with _ => _ end ] - => pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig (to_twisted P1)) _ _ (proj2_sig (to_twisted P2))) - end; t. - Qed. - - Global Instance isomorphic_commutative_group_m1 : - @Group.isomorphic_commutative_groups - Epoint E.eq - (E.add(nonzero_a:=nonzero_a)(square_a:=square_a)(nonsquare_d:=nonsquare_d)) - (E.zero(nonzero_a:=nonzero_a)) - (E.opp(nonzero_a:=nonzero_a)) - point eq m1add zero opp - from_twisted to_twisted. - Proof. - eapply Group.commutative_group_by_isomorphism; try exact _. - par: abstract - (cbv [to_twisted from_twisted zero opp m1add]; intros; - repeat match goal with - | |- context[E.add ?P ?Q] => - unique pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig P) _ _ (proj2_sig Q)) end; - t). - Qed. - End TwistMinusOne. - End ExtendedCoordinates. -End Extended. |